Metric Sobolev spaces I: equivalence of definitions
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910413458767872 |
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| author | Ambrosio, Luigi Ikonen, Toni Lučić, Danka Pasqualetto, Enrico |
| author_facet | Ambrosio, Luigi Ikonen, Toni Lučić, Danka Pasqualetto, Enrico |
| contents | This is the first of two works concerning the Sobolev calculus on metric measure spaces and its applications. In this work, we focus on several notions of metric Sobolev space and on their equivalence. More precisely, we give a systematic presentation of first-order $p$-Sobolev spaces, with $p\in[1,\infty)$, defined over a complete and separable metric space equipped with a boundedly-finite Borel measure. We focus on three different approaches: via approximation with Lipschitz functions; by studying the behaviour along curves, in terms either of the curve modulus or of test plans; via integration-by-parts, using Lipschitz derivations with divergence. Eventually, we show that all these approaches are fully equivalent. We emphasise that no doubling or Poincaré assumption is made, and that we allow also for the exponent $p=1$.
A substantial part of this work consists of a self-contained and partially-revisited exposition of known results, which are scattered across the existing literature, but it contains also several new results, mostly concerning the equivalence of metric Sobolev spaces for $p=1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_11190 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Metric Sobolev spaces I: equivalence of definitions Ambrosio, Luigi Ikonen, Toni Lučić, Danka Pasqualetto, Enrico Functional Analysis Analysis of PDEs Metric Geometry 49J52, 46E35, 53C23, 46N10, 46E15, 31C15, 28A12, 26A46 This is the first of two works concerning the Sobolev calculus on metric measure spaces and its applications. In this work, we focus on several notions of metric Sobolev space and on their equivalence. More precisely, we give a systematic presentation of first-order $p$-Sobolev spaces, with $p\in[1,\infty)$, defined over a complete and separable metric space equipped with a boundedly-finite Borel measure. We focus on three different approaches: via approximation with Lipschitz functions; by studying the behaviour along curves, in terms either of the curve modulus or of test plans; via integration-by-parts, using Lipschitz derivations with divergence. Eventually, we show that all these approaches are fully equivalent. We emphasise that no doubling or Poincaré assumption is made, and that we allow also for the exponent $p=1$. A substantial part of this work consists of a self-contained and partially-revisited exposition of known results, which are scattered across the existing literature, but it contains also several new results, mostly concerning the equivalence of metric Sobolev spaces for $p=1$. |
| title | Metric Sobolev spaces I: equivalence of definitions |
| topic | Functional Analysis Analysis of PDEs Metric Geometry 49J52, 46E35, 53C23, 46N10, 46E15, 31C15, 28A12, 26A46 |
| url | https://arxiv.org/abs/2404.11190 |